Mergelyan type theorems for some function spaces

Publicacions Matem`atiques, Vol 39 (1995), 61–69. MERGELYAN TYPE THEOREMS FOR SOME FUNCTION SPACES Arne Stray Abstract Let Fbe a relatively closed subset of the unit disc D.IfAis any of the Hardy spaces Hp(D), 0 <p<∞,A|Fdenotes the functions on Fbeing uniform limits of elements from Hp(D). Let ˜ Fconsist of all z∈Dsuch that |f(z)|≤sup{|f(z)|z∈F}for any bounded analytic function in D. It is proved that A|Fconsist of all functions fthat can be decomposed as f=u+v, where u belongs to Hp(D) and vis a uniformly continuous function on the set ˜ F, analytic at interior points of ˜ F. Let Abe a linear space of analytic functions and Fa subset of the complex plane Csuch that each f∈Ais defined on F. We denote by A|Fthe functions being uniformly approximable on Fby sequences from A. The aim with this paper is to give a partial solution to problem 8.5 no. 2 in [7]. If Ais any of the classical Hardy spaces Hp(D), 0 <p<∞, our main result is that A|Fcoincides (modulo the approximating space) with a well defined algebra of uniformly continuous analytic functions on F. Before giving a precise formulation of the main result, we need some definitions. Let Cua(F) denote the functions on Fbeing analytic in its interior F0and admitting continuous extension to the extended complex plane C∪{∞}.IfFis a compact subset of Cand Pconsists of the polynomials, a famous theorem of S. N. Mergelyan [7] can be formulated as P|F=Cua(˜ F) where ˜ Fis the union of Fand the bounded components of C/F. 62 A. Stray Suppose now we replace Pby the set H(C) consisting of all entire functions. Also allow Fto be a closed but possibly unbounded subset of C. Then it can be proved that (I): H(C)|F=H(C)+Cua(˜ F) where ˜ Fis the union of Fand certain components of C/F. A component Vis to be included in ˜ Fif and only if V∪ {∞} is not arcwise connected in C∪ {∞}. For details see [8], [9] and [10]. In general Amay contain unbounded functions. For this reason it is natural to look for an identity like (I) if we seek to describe A|Fin terms of uniformly continuous analytic functions. Let us use the notation gB= sup{|g(x)|:z∈B}if gis a function defined on the set B.We also define the hull of Fwith respect to A: ˆ FA={z:|f(z)|≤fF,f∈A}. We look for spaces Asatisfying the following: (∗): A|F=A+Cua(ˆ FA). Our main result is the (∗) is valid for the classical Hardy sapces HP(D) in the unit disc D,0<p<∞, when Fis any relatively closed subset of D. Also note that the two introductory examples are special cases of (∗). We refer to [3]or[5] for the basic theory of Hp(D), 0 <p≤∞. In particular H∞(D) denotes the bounded analytic functions in D.If F⊂Dis relatively closed, let ˆ F={z∈D:|f(z)|≤fF,f∈H∞(D)}. Our main result is Theorem 1. If 0<p<∞, then Hp(D)|F=Hp(D)+Cua(ˆ F). Proof of Theorem 1: If f∈Hp(D) it is easy to find {fn}⊂H∞(D) such that |fn(z)|≤|f(z)|and fn(z)→f(z) for z∈D. This shows that ˆ FA=ˆ Fif A=Hp(D), 0 <p<∞. Let us first prove that Hp(D)|F⊂Hp(D)+Cua(ˆ F). If g∈Hp(D)|F is bounded, we may assume g= n fn,f n∈Hp(D) Mergelyan type theorems for some function spaces 63 in the sense that nfnF<∞. There are two special classes of sets Fwhere a short proof of the decomposition of gcan be found. It may be instructive to consider these cases prior to the general proof. Let us firs assume that Fis a Farrell set for Hp(D). (See [8] for definition and various properties of these sets). Then we can find polynomials pn,n=1,2 such that pnF≤fnF+2 −n and pn−fnHp≤2−n for n=1,2,... . This gives a decomposition g= n (fn−pn)+ n pn|ˆ F as claimed. In our second example, we assume that Fcan be written as a Blaschke sequence S={ζν}, meaning that (1)  ν 1−|ζν|<∞. Then it is well known that the Blashcke product B(z)=Π |ζν| ζν ζν−z 1−¯ ζνz converge in D. Using cofinite subproducts Bnof B, we can obtain (1 −Bn)fnHp(D)<2−n,n=1,2,... and again we have a decomposition g= n (1 −Bn)fn+ n Bnfn|F. The geometric properties of the set Fare quite different in the two cases just discussed. To clarify this, let Fnt denote the non tangential closure of Fon the unit circle T.Soz∈Fnt if z∈Tand zis a limit of a sequence {zn}⊂Fsatisfying |z−zn|≤C(1 −|zn|), n=1,2,..., where Cmay depend only on z. We also define Ft=F∩T\Fnt. 64 A. Stray It is well known that Fis a Farrell set for Hp(D) if and only if the linear measure |Ft|of Ftis zero ([8]). On the other hand, the condition (1) is easily seen to imply that |Fnt|=0. We have thus obtained the decomposition of Hp(D)|Fin two rather different situations. The general proof will be divided into parts reflecting the “geometry” of the cases considered above. We shall argue as in the proof where Fwas a Farrell set, but the polynomials pnwill be replaced by functions from Hp(D) having a uniformly continuous restriction to F. The key part of the proof is an approximation argument related to the set Fnt: Lemma 1. Given f∈Hp(D),0<p<∞, and >0, there is an open set Vand f1∈Hp(D)with the following properties: (i) f−f1Hp(D)< (ii) f1F<fF+ (iii) f1extends continuously to F∩V (iv) |Fnt\V|=0. For the moment we take Lemma 1 for granted. Consider the “tangential” part Ftof F∩T. Let Kbe a compact subset of Ft. We assume there is a number δ=δ(K) such that Iz∩K=φif z∈F,|z|>1−δ, and Izdenotes the arc Iz={eiθ :|z−eiθ|≤2(|−|z|)}. By a construction due to J. Detraz ([2, Prop. 3.1]), we can find an outer function GK∈H∞(D) such that |GK|≤1 and GK(z)→0ifz→Kand z∈F. Moreover, GKextends to be continuous and non zero at any eiθ ∈T\K. We can find an increasing sequence of such sets Kn⊂Ft\Vwith corresponding outer functions Gn, such that |Ft\V\Kn|→0 and such that G= n Gnhas the following properties (i) 0 <|G(z)|≤1, z∈D (ii) Gextends to be continuous at any eiθ ∈V∩T. It also follows from the construction of {Gn}that G(z)→0ifz∈Fand z→z0∈∪ nKn. Consider finally the set L=( FT)\V\ n Kn. Mergelyan type theorems for some function spaces 65 It is evident that the linear meausre |L|of Lis zero. By a general version of the Rudin-Carleson theorem ([2]) there is H∈H∞(D) with continuous extension to L∪(T\L) such that H=0inDand H=0on L. For n=1,2,... we consider the functions Unin Hp(D) given by Un=G1 nH1 nf1 where f1satisfies the conclusions of Lemma 1. It follows from the construction of f1,Gand H, that Un|Fis uniformly continuous. This implies that Un|ˆ F∈Cua(ˆ F), by the maximum principle. To be a little bit more specific, suppose z0∈F∩Tand that Un(z)→0asz→z0and z∈F. Then |Un|<in F∩∆(z0), for some disc centered at z0. Choose a polynomial ppeaking at z0such that |Unp|<on F. Then if p(z0) = 1, we have lim sup z→z0 z∈ˆ F |Un(z)|= lim sup z→z0 z∈ˆ F |Un(z)p(z)|≤ since Unpˆ F≤UnpF≤. We turn to the proof of Theorem 1. If f∈Hp(D) and >0is given, we have shown (modulo proving Lemma 1) that there is a function U=Un∈Hp(D)∩Cua(ˆ F) with nso large that f−UHp(D)< U≤fF+. The proof of Theorem 1 now follows the introductory argument we gave in the special case where Fis a Farrell set. Let us finally prove Lemma 1. We may assume that fis bounded in D. So given f∈H∞(D), and >0, we consider a compact set K⊂Fnt. We shall require several properties of Krelated to f.If0<α<π, T(θ,α) denotes the cone in Dwith opening angle α, terminating at eiθ, and being symmetric with respect to the radius {reiθ,0≤r<1}.We assume that f(eiθ) = lim f(z) holds uniformly in eiθ ∈Kas z→eiθ inside T(θ,α). Now fix p∈(0,∞). Since f∈Hp(D), the radial limits f(eiθ), 0 ≤θ<2π, belong to Lp(dθ). We assume that Kis included in the Lebesgue set for fand that (3) 1 2rθ+r θ−r |f(eiϕ)−f(eiθ)|pdϕ →0 66 A. Stray uniformly in eiθ ∈Kas r→0. Such a set Kcan be found with |Fnt\K| as small we please. Fix δ>0 so small that |f(eiθ)−f(z)|<if eiθ ∈K,|z|>1−δ and z∈T(θ,α). We are now in a convenient position for applying Vitushkin’s scheme for approximation (see [13]or[4]). Let {∆j}N j=1 be a finite collection of open discs with centers zj∈Kand a common radius r<δ. Following Vitushkin’s scheme, let ϕj∈C1 0(∆j) be chosen such that 0 ≤ϕ≤1 and ϕ≡1in∆ 1 j=z:|z−zj|<r 2. As a preliminary approximation to fwe define fK=f−GK where GK= j Tϕj(f−f(zj))−rjis a finite sum which we shall explain in some detail. We assume fis defined outside of Dby f(z−1)=f(z). For general properties of the Tϕ-operator we refer to [11]or[4, page 30]. Here we only note that Tϕj(f−f(zj))(ς)=ϕj(ς)(f(ς)−f(zj)) −1 π∆jf(z)−f(zj) z−ς ∂ϕj ∂z dx dy(z) =Uj+Vjsay. We assume  ∂ϕj ∂z ≤A r, where Ais a numerical constant. Since f∈ Hp(D), we have in particular that f∈Lp(dx dy) locally. Therefore the convolution term Vjis continuous as a function of ς.Ifαis close to π, H¨olders inequality gives that |Vj(ς)|≤, ς ∈C j=1,2...N. Note also that Vjis analytic outside ∆j. According to Vitushkin’s scheme, the functions rjshould be analytic outside a compact subset of ∆j\Dand with the property that (Vj−rj)(ς) has a zero of order 3 at ∞. In addition we should require (4): rj∞≤A1Vj∞≤A1, j =1,2,... ,N where A1is a numerical constant. In our simple situation, the existence of {rj}is rather evident ([4, page 210–214]). From the individual bounds (3), it is part of Vitushkin’s scheme that (5):       N  j=1 (Vj−rj)     ∞ ≤A2 Mergelyan type theorems for some function spaces 67 for some numerical constant A2. We have not claimed that {∆1 j}N j=1 cover all of K. In fact we shall assume that ∆j∩∆k=φif j=k.In addition we assume that  K∩ N  1 ∆1 j ≥A3|K| for some numerical constant A3, where ∆1 j=z:|z−zj|≤r 2.We remark that fK=f−GKis analytic near ∆1 j∩Tfor 1 ≤j≤N. This is seen by writing fk=(f−Uj)−Vj− i=j (Ui+Vi)+ n  i=1 ri and inspecting these four terms separately. Note that the (Fatou) boundary values f(zj) satisfy |f(zj)|≤fF. Since fK=f1−ϕj+ j ϕjf(zj)+ j (Vj−rj) we have (6) fKF≤fF+A2. From (3) and (5) we also get f−fKHp(D)=GKHp(D)≤(1 + A2) if ris sufficiently small. The function fKsatisfies the conditions for f1in Lemma 1 except that fKis only analytic (and hence continuous) near a subset PKof K. But since |PK|≥A3|K|, Lemma 1 follows by repeating our construction countably many times. The main reason why repetition works, is that the Tϕ-operator preserves continuity and analyticity ([4, page 30]). It remains to show that Cua(ˆ F)⊂Hp(D)|ˆ F. Let Bdenote the Banach algebra H∞(D)|ˆ F. Also put X=(ˆ F). If Vis a component of C\X, there mus exist h∈H∞(D) such that 1=h(z0)>hF for some z0∈V. But then 1 −his invertible in Band since 1−h=(z−z0)g, g ∈H∞(D) 68 A. Stray we conclude that (z−z0)−1|ˆ F∈B. This means that R(X)|ˆ F⊂B, where R(X) is the uniform closure on Xby the rational functions with poles off X. But if {Vj}are the components os C\X, the maximum principle gives ∂Vj∩T=φ,j=1,2,... and hence ∂X =∪∞ 1∂Vj. For such sets X(with empty “inner boundary”) Vitushkin has proved that R(X)=Cua(X) ([4, page 219]), and hence Theorem 1 is proved. This solves completely problem 8.5 no. 2 in [7] for the space Hp(D), 0<p<∞.Forp=∞the problem is still open. For p=∞, some information about H∞|Fcan be obtained from the work by Carl Sundberg in [12]. If f∈BMOA and f|Fis bounded, Sundberg shows that f∈H∞|F. On the other hand, our proof above shows that any f∈H∞|Fcan be written as f=u+vwith u∈H∞ and v∈∩ p>0Hp(D). Several questions arises from this. Here we only mention the following: Let f∈BMOA be bounded on a relatively closed set F⊂D. Is there g∈H∞such that the restriction (f−g)|Fis uniformly continuous on F? References 1. A. M. Davie and A. Stray, Interpolation sets for analytic functions, Pacific J. Math. 42(1) (1972). 2. J. Detraz, Algebres de fonctions analytiques dans le disque, Ann. Sci. Ecole Norm. Sup. 4e serie (1970), 313–352. 3. P. L. Duren,“Theory of Hpspaces,” Academic Press, 1970. 4. T. W. Gamelin,“Uniform Algebras,” Prentice Hall, Englewood Cliffs, N. J., 1969. 5. J. Garnett,“Bounded Analytic Functions,” Academic Press, 1980. 6. S. N. Mergelyan, Uniform approximation to functions of a complex variable, Urephi Mat. Nauk. 7(2) (1952), 31–122. 7. ?,“Linear and Complex Analysis Problem Book,” Lecture Notes in Mathematics 1043, Springer Verlag, 1984. 8. F. Perez-Gonzalez and A. Stray, Farrell and Mergelyan sets for Hp-spaces, 0 <p<1, Michigan Math. J. 36 (1989), 379–386. 9. A. Stray, Decomposition of approximable functions, Ann. of Math. 120 (1984), 225–235. 10. A. Stray, Approximation by analytic functions which are uniformly continuous on a subset of their domain of definition, American J. Math. 99 (1977), 787–800. Mergelyan type theorems for some function spaces 69 11. A. Stray, Characterization of Mergelyan sets, Proc. Amer. Math. Soc. 44 (1974), 347–352. 12. C. Sundberg, Truncations of BMO functions, Indiana University Math. I. 33(5) (1984), 749–779. 13. A. G. Vitushkin, The analytic capacity of sets and problems in approximation theory, Russian Math. Surveys 22 (1967), 139–200. Department of Mathematics University of Bergen All´egt 55 5007 Bergen NORWAY Primera versi´o rebuda el 2 de Setembre de 1993, darrera versi´o rebuda el 9 de Febrer de 1995